Theorems · Theorem · general topology
Metric.ball_mem_nhds
∀ {α : Type u} [inst : PseudoMetricSpace α] (x : α) {ε : ℝ}, 0 < ε → Metric.ball x ε ∈ nhds x- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 64 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Filterstatement · cited by 8,121
- nhdsstatement · cited by 5,554
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.ballstatement · cited by 735
- IsOpen.mem_nhdsproof · cited by 470
- Metric.isOpen_ballproof · cited by 63
- Metric.mem_ball_selfproof · cited by 40
Cited by64
Results whose statement or proof uses this declaration.
- Metric.closedBall_mem_nhdsproof · cited by 20
- Complex.hasDerivAt_expproof · cited by 10
- NormedSpace.isVonNBounded_iffproof · cited by 9
- Metric.disjoint_nhds_coboundedproof · cited by 7
- hasFDerivAt_integral_of_dominated_of_fderiv_leproof · cited by 5
- VectorFourier.hasFDerivAt_fourierIntegralproof · cited by 5
- summable_geometric_iff_norm_lt_oneproof · cited by 4
- Asymptotics.isLittleOTVS_oneproof · cited by 4
- hasDerivAt_integral_of_dominated_loc_of_deriv_leproof · cited by 4
- hasFDerivAt_integral_of_dominated_loc_of_lipproof · cited by 4
- continuousAt_of_locally_lipschitzproof · cited by 3
- ProbabilityTheory.hasDerivAt_integral_pow_mul_expproof · cited by 3